# Full text of "Treatise On Analysis Vol-Ii"

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```114       XIII    INTEGRATION
(1 3.5.2.1 )                    n*(/)

n                            n->oo

Suppose first of all that/e <2fR(X) and that/, e jfR(X) for all n. It is
clear that the supports of all the fn are contained in the same compact set
K = Supp(/) u Supp(/i). By Dini's theorem (7.2.2), the sequence (/) con-
verges uniformly to /on X. The formula (13.5.2.1) then follows from the fact
that the restriction of n to Jf(X;K) is a continuous linear form on this
Banach space.

Now pass to the general case. Clearly we have ju*(/J ^ /**(/) for all n.
Hence it is enough to show that, for each function u e jf R(X) such that
u ^/ we have ju*(w) ^ sup /**(/). Now by (12.7.8) we know that, for each
function/,, there exists an increasing sequence (gmn)m^i of functions belong-
ing to 3ffR(X) such that / = sup gmn . We have /= sup gmn, hence also

m                                                        m, n

/= sup hn , where hn =    sup   #M . The functions hn clearly belong to *?f R(X)

n                             P^n,q^n

and form an increasing sequence. Since u ^/, we have u = sup(inf(w, An)); the

n

sequence of functions inf(w, hn) is increasing and belongs to Jf R(X); since also
u eĞ2fR(X), the first part of the proof shows that /i*(w) = sup ju*(inf(w, /*));

n

but since hn ^/,  we have /^*(inf(w, /*)) ^ M*(/M),  and therefore  finally

Q.E.D.

We remark that because the functions in </ never take the value  oo,
the sum/ +/2 of two such functions is defined at every point of X, and
belongs to J (12.7.5).

(13.5.3)   7//i,/2 e J, then ^(f, +/2) = /.*(/) +

We may write / = lim gn and /2 == lim ^w , where (gn) and (/ZM) are two

increasing sequences of functions belonging to ^TR(X) (12.7.8). Hence we
have / +/2 = lim (gn -f /?)   (4.1.8).   Since  ]n(gn + hn) = /*(#) -f ju(^i)>  the

result follows from (13.5.2) and (4.1.8).

Let (tn) be any sequence whose terms are real numbers ^0, or +00.
Since the partial sums \$ = /t +    -f tn are defined (4.1.8) and form an

___                                                                            00

increasing sequence, this sequence has a limit in R (4.2.1), denoted by J] tn
and called the sum of the series with general term tn. For every sequence

00

(/,) of positive functions belonging to I, the function x\> ^fn(x) is therefore

n = locally compact spaces and TT : X -> Y a proper continuous mapping
```